and the Laws of Plane Waves propagated through them. 125 



These are the equations of motion; also equations (0) will become 



(dVJ, _ dV^\ clF _ (dV^ _ clVl\ (IF _ 

 dZ dZ ) dy \dY dY j dz ~ ' 



dn_dVl\(}P_(<^m_dVl\dF_ 



dX dX ) dz \dZ dZ ) dx~ ' ^-^^ 



dV, dV: 

 dY dY 



\ dF (dV, dV'„'\dF_ 

 ) dx \dX dx) dy ~ 



These are the conditions to be fulfilled at the limiting surface F {x,y,z) =0. 

 They are equivalent to two conditions only, as the third equation may be de- 

 duced from the first two. The reason of this is evident a priori; for the con- 

 ditions at the limits express in general, that the normal and two tangential 

 pressures arising from the molecular forces in each body equilibrate each other 

 for every point of the separating surface ; but in the present case there are no 

 normal pressures ; hence there can be only two mechanical conditions at the 

 limits. To these conditions at the limits, arising from the mechanical equa- 

 tions, should be added the three geometrical conditions resulting from the equi- 

 valence of vibrations ; these conditions are 



?0 = ?0 ) Vo = Vo 1 S ~ b u • 



These, together with the mechanical conditions, will be equivalent to five equa- 

 tions, which Professor Mac Ccllagh reduces to four by the hypothesis that the 

 density of the kxminiferous medium is the same in all transparent bodies ; this 

 hypothesis is necessary in order to reduce the number of equations to the num- 

 ber of unknown quantities in the problem. 



If no external forces act upon the system, the function V will be reduced 

 to a homogeneous function of the second order, 



2 F = PT- -t- Q }" -f BZ' + 2FYZ + 2GXZ + 2HXY; 



and since (X, Y, Z) are transformed by a change of co-ordinates, in the same 

 manner as {x, y, z), it is evident that the coefiicients of the rectangles will 

 vanish for axes of co-ordinates which coincide with the axes of the elUpsoid, 



Px- + Qy- -t- Pz- -f 2Fyz + 2Gxz -)- 2Hxy = 1. 



